Method and apparatus for encoding and decoding data

ABSTRACT

A method and apparatus for selecting interleaver sizes for turbo codes is provided herein. During operation information block of size K is received. An interleaver size K′ is determined that is related to K″, where K″ from a set of sizes; wherein the set of sizes comprise K″=a p ×f, p min ≦p≦p max ; f min ≦f≦f max , wherein a is an integer and f is a continuous integer between f min  and f max , p takes integer values between p min  and p max , a&gt;1, p max &gt;p min , p min &gt;1. The information block of size K is padded into an input block of size K′ using filler bits, if needed. Encoding is performed using the original input block and the interleaved input block to obtain a codeword block using a turbo encoder. The codeword block is transmitted through the channel.

FIELD OF THE INVENTION

The present invention relates generally to encoding and decoding data and in particular, to a method and apparatus for turbo encoding and decoding.

BACKGROUND OF THE INVENTION

Digital data transmissions over wired and wireless links may be corrupted, for instance, by noise in the link or channel, by interference from other transmissions, or by other environmental factors. To combat the errors introduced by the channel, many communication systems employ error-correction techniques to aid in communication.

One technique utilized for error correction is turbo coding of an information block before it is transmitted over the channel. Utilizing such a technique, an encoder within the transmitter of a communication system will encode an input block u of length K′ bits into a codeword block x of N bits. The codeword block is then transmitted over the channel, possibly after further processing such as channel interleaving as defined in the IEEE 802.16e specifications. At the receiver, the turbo decoder takes the received signal vector y of length N as input, and generates an estimate û of vector u.

Typically the turbo encoder is composed of two constituent convolutional encoders. The first constituent encoder takes the input block u as input in its original order, and the second constituent encoder takes the input block u in its interleaved order after passing u through a turbo interleaver n. The turbo encoder output x is composed of the systematic bits (equal to the input block u), the parity bits from the first constituent encoder, and the parity bits from the second constituent encoder.

Correspondingly the turbo decoder within the receiver of the communication system is composed of two constituent convolutional decoders, one for each constituent code. The constituent decoders are separated by the interleaver π and the corresponding deinterleaver π⁻¹. Messages in the format of log-likelihood ratios (LLRs) are passed between the constituent decoders iteratively. The decision û is made after several iterations.

The turbo interleaver π is the key component in the turbo code design. It is responsible for scrambling the input block u in a pseudo-random fashion, thus providing the codewords x with good weight distribution, hence good error-correcting capabilities. In addition to decoding performance, the definition of the turbo interleaver π greatly impacts the implementation of the turbo decoder within the receiver. To allow high-level of parallel processing without memory access contentions, the turbo interleaver π needs to have contention-free properties.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a transmitter.

FIG. 2 is a block diagram of the turbo encoder of FIG. 1.

FIG. 3 is a block diagram of a receiver.

FIG. 4 is a block diagram of the turbo decoder of FIG. 4.

FIG. 5 is a flow chart showing operation of the transmitter of FIG. 1.

FIG. 6 is a flow chart showing operation of the receiver of FIG. 3.

DETAILED DESCRIPTION OF THE DRAWINGS

In order to address the above-mentioned need for contention-free interleavers, a method and apparatus for selecting interleaver sizes for turbo codes is provided herein.

During operation an information block of size K is received. An interleaver size K′ is determined where K′ is related to K″ where K″ is from a set of sizes; wherein the set of sizes comprise K″=a^(p)×f, p_(min)≦p≦p_(max); f_(min)≦f≦f_(max), wherein a is an integer, f is a continuous integer between f_(min) and f_(max), and p takes integer values between p_(min) and p_(max), a>1, p_(max)>p_(min), p_(min)>1. The information block of size K is padded into an input block of size K′. The input block is interleaved using an interleaver of size K′. The original input block and the interleaved input block are encoded to obtain a codeword block. The codeword block is transmitted trough the channel.

In a further embodiment of the present invention the step of determining the interleaver size K′ that is related to K″ comprise the step of using K′=K″.

In yet another embodiment of the present invention the step of determining the interleaver size K′ that is related to K″ comprise the step of using K′=K″ when K″ is not a multiple of (2^(m)−1); otherwise using K′=K″+δ(K″) when K″ is a multiple of (2^(m)−1), wherein m is the memory length of the constituent convolutional encoder, and δ(K″) is a small positive or negative integer not equal to a multiple of (2^(m−)1). In one embodiment m=3.

In yet another embodiment of the present invention the step of interleaving the input block comprises the step of using a permutation π(i)=(i P₀+A+d(i))mod K′, where 0≦i≦K′−1 is the sequential index of the symbol positions after interleaving, π(i) is the symbol index before interleaving corresponding to position i, K′ is the interleaver size in symbols, P₀ is a number that is relatively prime to K′, A is a constant, C is a small number that divides K′, and d(i) is a dither vector of the form d(i)=α(i mod C)+P₀×β(i mod C) where α(·) and β(·) are vectors each of length C, periodically applied for 0≦i≦K′−1.

Prior to describing encoding and decoding data, the following definitions are provided to set the necessary background:

-   -   K denotes the size of an information block.     -   K′ denotes an interleaver size (i.e., input block size for which         a turbo code interleaver is defined).     -   K″ denotes an auxiliary variable that may be used in determining         an interleaver size.     -   K_(filler) denotes the number of filler bits added to the         information block.     -   π denotes the turbo code internal interleaver.     -   The flooring operation └x┘ denotes the largest integer smaller         than or equal to x and the ceiling operation ┌x┐ denotes the         smallest integer larger than or equal to x.     -   u denotes an input block, which has a length of K′ and is sent         to the turbo encoder at the transmitter. û0 denotes the         estimated input block, which has a length of K′ and is produced         by the turbo decoder at the receiver. Note tat û=u when there is         no decoding error. Otherwise û≠u.

Turning now to the drawings, wherein like numerals designate like components, FIG. 1 is a block diagram of transmitter 100. As shown, transmitter 100 comprises filler insertion circuitry 109, turbo encoder 101, interleaver size determination circuitry 103, interleaver parameter table 105, and transmitter 107. Encoder 101 is preferably a rate-⅓ 3GPP turbo coder, however, the techniques described herein for operating encoder 101 may be applied to other encoders, including, but not limited to turbo coders performing turbo coding with tail bits or no tail bits, tail-biting, binary or duo-binary turbo coders, turbo coders using different rate-matching and puncturing techniques, . . . , etc. Circuitry 103 determines the interleaver size K′ that is related to K″, where K″ is from a set of sizes; wherein the set of sizes comprise K″=a^(p)×f, p_(min)≦p≦p_(max); f_(min)≦f≦f_(max), wherein a is an integer, f is a continuous integer between f_(min) and f_(max) and p takes integer values between p_(min) and p_(max), a>1, p_(max)>p_(min), p_(min)>1.

During operation of transmitter 100, information block of size K needs to be encoded by the turbo encoder 101. For some communication systems where a large number of different Ks are used, it is not efficient (and often impossible) to define a contention-free (CF) interleaver for every information block size K. It is preferable if a small set (K′) of well-designed CF interleavers is able to cover all the information block sizes. Given an information block size K, a suitable interleaver size K′ may be chosen by circuitry 103 from the set of available sizes (e.g., interleaver sizes listed in table 105). The information block is then padded into an input block of size K′ by circuitry 109 and sent as input to the turbo encoder 101. A typical arrangement is to pad the information block with K_(filler) filler bits (via filler insertion circuitry 109). Note that the term “size” and “length” are used interchangeably to indicate the number of elements in a block or vector.

Once K′ is chosen by circuitry 103, it is provided to turbo encoder 101. During encoding, a contention-free interleaver may be used (not shown in FIG. 1). For instance, the interleaver can use a permutation π(i)=(iP₀+A+d(i)) mod K′, wherein 0≦i≦K′−1 is the sequential index of the symbol positions after interleaving, π(i) is the symbol index before interleaving corresponding to position i, K′ is the interleaves size in symbols, P₀ is a number that is relatively prime to K′, A is a constant, C is a small number that divides K′, and d(i) is a “dither” vector of the from d(i)=α(i mod C)+P₀×β(i mod C) where α(·) ad β(·) are vectors each of length C, periodically applied for 0≦i≦K′−1. In general a symbol may be composed of multiple bits and the step of interleaving may use an additional step of permuting the bits within a symbol. Without losing generality, the discussion below considers the typical case where a symbol is composed of one bit only (thus no need of permuting bits within a symbol), and the terms “bit” and “symbol” can be used interchangeably.

The output of turbo encoder 101 comprises a codeword block x, and x is sent to transmitter 107 where it is transmitted through the channel. The transmitter may perform additional processing such as rate matching, channel interleaving, modulation, etc., before transmitting the codeword block x through the channel.

FIG. 2 is a block diagram of encoder 101 of FIG. 1. As shown, encoder 101 comprises interleaver 201, encoding circuitry 202, and encoding circuitry 203. An example of the encoder is the turbo coder defined in 3GPP specification. The mother code rate of the turbo coder defined in 3GPP has a native code rate of R=⅓. At the output of the turbo encoder, three bits are produced for each bit within the input block: one systematic bit (equal to the bit in the input block), one parity bit from constituent encoder 1, one parity bit from constituent encoder 2. In addition, the output of the turbo encoder may also include N_(TB) tail bits, which are used to terminate the trellis of the constituent codes. For example, for 3GPP turbo code, N_(TB)=12 bits at the output of the turbo encoder, 6 tail bits per constituent code. On the other hand, it is possible to use tail-biting constituent convolutional codes, thus making N_(TB)=0.

Interleaver 201 can be a contention-free interleaver. An interleaver π(i), 0≦i<K′, is said to be contention-free for a window size W if and only if it satisfies the following constraint for both ψ=π (interleaver) and ψ=π⁻¹ (de-interleaver),

$\begin{matrix} {\left\lfloor \frac{\phi \left( {j + {t\; W}} \right)}{W} \right\rfloor \neq \left\lfloor \frac{\phi \left( {j + {v\; W}} \right)}{W} \right\rfloor} & (1) \end{matrix}$

where 0≦j<W, 0≦t; ν<M(=K′/W), and t≠ν. Though it is not always necessary, for efficient turbo decoder design, typically all the M windows are full, where K′=MW. The terms in (1) are the memory ban addresses that are concurrently accessed by the M processors when writing the extrinsic values to the output memory bans during iterative decoding. If these memory bank addresses are all unique during each read and write operations, there are no contentions in memory access and hence the (de)interleaving latency can be avoided, leading to a high speed decoder implementation.

During operation of turbo encoder 101, input block of length K′ bits enters both interleaver 201 and encoding circuitry 202. Interleaver 201 can be a contention-free interleaver of size K′.

Interleaver 201 interleaves the input block and passes the input block in interleaved order to encoding circuitry 203. Encoding circuitry 203 then encodes the interleaved input block. In a similar manner, encoding circuitry 202 encodes the original input block. The codeword block x is composed of systematic block (equal to the input block), output of encoding circuitry 202, and output of encoding circuitry 203. The codeword block x is then sent to transmitter 107 which can also receive a copy of the input block directly.

As an example of the contention-free interleaver, an almost regular permutation (ARP) interleaver is given by the following expression

π(i)=(iP ₀ +A+d(i))mod K′

where 0≦i≦K′−1 is the sequential index of the bit positions after interleaving, π(i) is the bit index before interleaving corresponding to position i, K′ is the interleaver size, P₀ is a number that is relatively prime to K′, A is a constant, C is a small number that divides K′, and d(i) is a dither vector of the form d(i)=α(i mod C)+P₀×β(i mod C) where α(·) and β(·) are vectors each of length C, periodically applied for 0≦i≦K′−1. Both α(·) and β (·) are composed of multiples of C. The overall interleaver π(·) thus constructed has quasi-cyclic (i.e., periodic) properties with period C, and when Used in tail-biting turbo codes, the turbo code itself becomes quasi-cyclic leading to a simplified code design procedure.

If interleaver 201 can satisfy (1) for various values of M, then the decoder can be implemented using various degrees of parallelism (one for each M). Thus it is desirable to choose K′ that has various factors. For an A interleaver of length K′, any window size W where W is a multiple of C and a factor of K′, can be used for high-speed decoding without memory access contentions. This provides flexibility and scalability in decoder design by allowing a wide range of parallelism factors M. Thus, a good compromise between decoding speed and complexity can be made based on system (or classes of user elements) requirements.

Choosing Interleaver Size K′:

As discussed above, interleaver size determination circuitry 103 needs to determine an interleaver size K′ for a given K, This section describes a way of selecting a limited number of sizes (i.e., K′) for which turbo code interleavers may be defined. As indicated previously, filler insertion circuitry (along with puncturing or rate-matching methods) may be used to handle any information block size K. In general, the interleaver size selection must take into consideration the decoding burden and performance degradation due to the filler bits.

The number of filler bits K_(filler) padded to an information block to form an input block is desirable to be limited to a small percent (e.g., around 10-13%) of the information block size K. This is achieved by limiting the difference between adjacent interleaver sizes, i.e., adjacent K′ values (assuming all available K′ values are sorted in ascending order). The number of filler bits are minimized by choosing the smallest K′ available such that K′≧K. The number of filler bits is K_(filler)=K′−K. However, other available values of K′≧K may also be chosen, if desired.

Consider the following set of sizes defined to cover information sizes between K_(min) and K_(max).

K″=a ^(p) ×f, p _(min) ≦p≦p _(max) ; f _(min) ≦f≦f _(max),   (2)

where a is an integer, f is a continuous integer between f_(min) and f_(max), and p takes integer values between p_(min) and p_(max), a>1, p_(max)>p_(min), p_(min)>1. Although not necessary, one can choose these parameters such that K_(min)=a^(pmin)×f_(min), and K_(max)=a^(pmax)×f_(max), while discarding any sizes that may not be needed. This method of selecting a limited set of sizes to cover a rage of information block sizes is referred to as semi-log slicing. For a given information block of size K, a size K′ related to a K″ based on the semilog-slicing table, and input block size K.

The semilog slicing is similar to the companding operation employed in compressing signals of large dynamic range, for example, A-law and mu-Law companders used in speech codecs. The semilog slicing rule allows an efficient design to cover a wide-range of information block sizes.

Of the several ways of choosing the parameters, one way of choosing f_(min) and f_(max) values is to let K″ values resulting from adjacent p line up with each other, i.e., a^(p)×(f_(max)+1)=a^(p+1)×f_(min), thus

ti f _(max) =a×f _(min)−1 For a given value of p, the separation between two adjacent block sizes K″ is given by a^(p), which means that a maximum of a^(p)−1 filler bits are added if the information block size K is in group p and the interleaver size is equal to K″. Thus, the fraction of filler bits K_(filler) over the information block size K is bounded as shown below, which occurs when the block size K is slightly greater than the size given by (p, f_(min)), and using K′=K″ given by p,f_(min)+1) to,

$\begin{matrix} {{\max \left( \frac{K_{filler}}{K} \right)} = \frac{a^{p} - 1}{{a^{p} \times f_{\min}} + 1}} \\ {\leq \frac{1}{f_{\min}}} \\ {= \frac{a}{\left( {f_{\max} + 1} \right)}} \end{matrix}.$

Alternatively, K″ values resulting from adjacent p can line up with each other via a^(p)×f_(max)=a^(p+1)x(f_(min)−1), resulting in f_(max)=a×(f_(min)−1). This would give a similar K_(filler)/K bound. Therefore, the parameters for the semi-log slicing can be tuned according to the range of block sizes to be supported, and also on the tolerable fraction of filler bits. The choice of f_(min) requires a balance between the following two requirements:

-   -   f_(min) should be large to reduce the fraction of the filler         bits;     -   f_(min) should be small to limit the size of the interleaver         table, since the number of block sizes defined for each p is         f_(max)−f_(min)+1=(a−1)×f_(min), assuming f_(max)=a×f_(min)−1.

The semi-log slicing method is very simple in that for any block size, the interleaver size K′ to be used may be easily determined based on a K″ computed from (2). Once the semilog slice sizes are defined (K″), the interleaver size K′ may be obtained from the semilog slice sizes (without deviating subtantially) by, for example,

-   -   1. Using K′=K″. In other words, the semilog slice sizes may be         used as valid interleaver sizes directly.     -   2. Using K′=K″ when K″ is not a multiple of (2^(m)−1), otherwise         using K′=K″+δ(K″) when K″ is a multiple of (2^(m)−1), wherein m         is a memory length of a constituent convolutional encoder, and         δ(K″) is a small positive or negative integer not equal to a         multiple of (2^(m)−1). This is helpful if the constituent         convolutional codes are tail-biting, where multiples of         (2^(m)−1) are invalid. The sizes defined by the semi-log slicing         method of (2) may sometimes include sizes that are unsuitable         interleaver sizes for turbo coding. For example, the tail-biting         version of the eight-state 3GPP turbo encoder (m=3) does not         support input block sizes (i.e., interleaver sizes) that are         multiples of 7 (i.e., 2^(m)−1). In such cases, whenever         equation (2) results in a size that is multiple of 2^(m)−1, a         small value is subtracted or added to it so that the resulting         size is no longer a multiple of 2^(m)−1.

For example, if a=2, f_(min)=8, and f_(max)=15, then interleaver sizes of the form K′=K″=2^(p)×14 are multiples of 7, and hence are invalid interleaver sizes when using tail-biting 3GPP TC. Therefore, this case must be handled with slight alteration, e.g., using K′=K″ when K″ is not a multiple of 7; otherwise using K′=K″+δ(K″) when K″ is a multiple of 7, and δ(K″) is a small positive or negative integer not equal to a multiple of 7.

For the K″ sizes that are invalid choice for ARP interleavers, one simple way to determine a related interleaver size K′ is by subtracting (addition is just as valid) d×C from K″, where d(is a small positive integer and d(is not a multiple of 7, and C is a AC interleaver cycle length used for the block sizes next to K′ in the set of available sizes. (Recall that the block size of an ARP interleaver is a multiple of the cycle lengths C.) In other words,

K′=K″−dC   (3)

or

K′=K″+dC   (4)

when K″ is a multiple of 7. Since C is normally an even integer, such as, 4, 8, 12, or 16, this adjustment gives two advantages, namely, (a) K′ is not a multiple of 7, and (b) K′ is a multiple of C and hence an ARP interleaver for size K′ can be designed.

For simplicity, the same d can be chosen for all K″ that need to be adjusted. One important consideration for choosing d is that it should be such that all sizes obtained by (3) or (4) have a substantial number of factors, which allows supporting a wide range of parallelism for the CF interleaver thus defined.

Example of ARP Interleaver Size Selection:

A set of CF ARP interleavers suitable to cover information block sizes for 3GPP Long Term Evolution (LTE) is shown in Table 1. The interleaver sizes available in table 105 are defined based on the semi-log slicing method described above. Specifically,

K″=2^(p) ×f, p=4,5, . . . , 9; f=8,9, . . . , 15,   (5)

and K′ is determined from K″. The interleaver sizes are determined as follows: using K′=K″ and for p=4,5,6,7,8,9 and f=8,9,10,11,12,13,15, and using K′=K″−dC for p=4,5,6,7,8,9 and f=14, covering K from 128 to 7680. The last thee sizes (f=13, 14, 15) corresponding to p=9 may be removed such that K_(max)=6144, with K_(min)=128. Equation (3) is used along with d=2 when f=14 (i.e., to avoid interleaver sizes that are multiples of 7) in order to handle tail-biting TC. These sizes are highlighted in Table 1. Once the interleaver sizes in 105 are determined, a CF interleaver may be designed for each interleaver size.

Given any information block size K, the circuitry 103 can determine the interleaver size K′ to be used for K by choosing the smallest value of K′ from 105 that is greater than or equal to K. With K known, and f_(min)=2^(b), f_(max)=2^(b+1)−1, where b is an integer, the parameters p and f can be calculated as follows,

$\begin{matrix} {p = {\left\lfloor {\log_{2}(K)} \right\rfloor - b}} & (6) \\ {f = \left\lceil \frac{K}{2^{p}} \right\rceil} & (7) \end{matrix}$

In particular, for the parameters in (5), b=3, and

p=└log₂(K)┘−3   (8)

With the parameters p and f the block size K′ can be calculated using (2) or (5), and moreover, when f is a multiple of 7 and tail-biting encoding is used, interleaver size calculated using (3) or (4) may be used in addition. The parameters associated with the interleaver of size K′ is then looked up from the storage means for interleaver parameter 105, which is normally stored in memory for the communication device.

Storage means for interleaver parameter 105 may store ARP interleaver parameters using the values of K′, C, P₀, α(·) and β (·) that are taken from at least one row of Table 1. The interleaver 201 may use an ARP interleaver with the values of K′, C, P₀, α(·) and β (·) that are taken from at least one row of the following table:

TABLE 1 A table of interleaver parameters for ARP interleavers with A = 3. Interleaver size K′ C P₀ α β 128 4 81 4 0 0 4 0 20 120 68 144 4 91 4 0 4 0 0 76 20 52 160 4 123 4 4 0 0 0 4 8 12 176 4 127 4 4 0 0 0 12 112 44 192 4 169 4 0 4 0 0 8 4 16 208 4 37 4 0 4 0 0 68 20 164 216 4 121 4 4 0 0 0 68 12 28 240 4 161 0 0 4 4 0 16 196 212 256 4 31 0 0 4 4 0 60 8 68 288 4 131 0 4 0 4 0 80 144 36 320 4 69 0 0 4 4 0 4 8 12 352 4 35 4 0 0 4 0 48 96 152 384 4 91 0 0 4 4 0 4 20 24 416 4 31 4 4 0 0 0 24 28 60 440 4 53 0 4 4 0 0 4 20 216 480 4 53 0 4 4 0 0 72 192 12 512 4 273 0 0 4 4 0 20 8 24 576 4 29 0 0 4 4 0 64 120 68 640 4 147 0 0 4 4 0 24 12 4 704 4 309 0 0 4 4 0 4 12 8 768 4 241 0 0 4 4 0 4 12 8 832 4 53 0 0 4 4 0 4 12 8 888 4 77 0 4 4 0 0 48 64 140 960 4 143 0 0 4 4 0 4 12 8 1024 8 245 8 0 8 8 0 0 8 0 0 8 40 16 96 80 56 88 1152 8 119 0 0 8 0 8 0 8 8 0 8 40 64 80 48 24 88 1280 8 897 0 0 8 0 8 0 8 8 0 8 96 88 32 16 48 40 1408 8 593 0 8 0 8 8 0 8 0 0 8 96 48 32 16 80 40 1536 8 1139 0 0 8 0 8 0 8 8 0 16 56 88 80 24 72 64 1664 8 1451 0 8 0 8 8 0 8 0 0 16 40 96 88 80 32 48 1776 8 115 8 0 0 0 0 8 8 8 0 88 56 40 152 120 128 200 1920 8 233 8 0 8 8 8 0 0 0 0 16 24 88 64 8 32 40 2048 8 77 0 0 8 8 0 8 8 0 0 64 136 160 48 192 24 120 2304 8 1631 0 0 8 0 8 0 8 8 0 24 80 40 16 96 64 32 2560 8 2249 0 0 8 0 8 0 8 8 0 8 72 40 88 48 32 96 2816 8 1235 8 8 0 0 0 8 0 8 0 16 88 96 56 24 48 64 3072 8 671 0 0 8 0 8 0 8 8 0 8 48 32 64 88 40 56 3328 8 1459 0 0 8 0 8 0 8 8 0 32 8 56 80 16 72 48 3568 8 147 0 0 8 0 8 0 8 8 0 72 64 48 88 8 184 248 3840 8 3721 8 0 8 8 8 0 0 0 0 16 48 24 8 32 40 88 4096 8 83 8 8 0 0 0 8 8 0 0 16 120 152 24 216 64 240 4608 8 181 0 8 0 8 0 8 8 0 0 32 176 216 136 64 224 248 5120 8 3629 0 8 0 8 8 0 8 0 0 16 40 96 88 80 32 48 5632 8 211 0 0 8 0 8 0 8 8 0 24 208 112 224 168 184 48 6144 8 4355 8 0 8 8 8 0 0 0 0 8 16 64 24 48 80 32 Note in the table, the dither vectors α (and similarly β) is written such that the leftmost value corresponds to α(0) (and similarly β(0)).

Properties of the ARP Interleaver:

There are several ways to modify the interleaver table. For example, storage can be reduced by using a set of ARP parameters that apply to more than one interleaver size. For example, the 1024-bit, 2048-bit, 4096-bit interleavers can all use the same ARP parameters. In another variation, some of the rows of the table may be redesigned based on different C values, if needed. I n another enhancement, some of the entries of the parameters (e.g., α(0) and β(0)) may be fixed (e.g., always zero).

Following are some further comments on the interleaver selection procedure used to obtain Table 1.

-   -   1. An Offset value A=3 is chosen for reducing storage.     -   2. Based on performance study and storage, a cycle length C=4 is         used for K′<1024, C=8 for K′≧1024.     -   3. For each block size, simulations were performed to make sure         the ARP interleaver performance (with tail-biting encoding) is         near to or better than the performance with the interleaver         defined in the specification for 3GPP Turbo code.     -   4. Table 1, based on (5) was defined to cover a particular set         of interleaver sizes (e.g., 128˜6144 bits). If preferred, other         interleaver sizes can be deleted or added.     -   5. All the interleavers defined in 105 can be used for either         tailed or tail-biting turbo codes depending upon the permissible         performance degradation.

FIG. 3 is a block diagram of the receiver 300. At the input, the filler handling circuitry 302 receives a signal vector which may have been transmitted through the channel, e.g., over-the-air. Circuitry 306 then determines the interleaver size K′ which can be done in a similar fashion as discussed above, for example by performing a table lookup from storage 308, or via calculations such as (7), (8) and (2). Therefore, given the information block size K, the decoder 304 uses the same interleaver size K′ that was used by the encoder 101. Filler handling circuitry 302 is used to appropriately process received signal vector and the filler bits positions (e.g., if filler bit positions are known, then the corresponding LLR magnitudes can be set to very high magnitudes during decoding). Then turbo decoder 304 performs decoding and obtains an estimate û of the input block of length K′. Finally the information block extraction circuitry 310 extracts an estimated information block from û. Though the filler handling circuitry 302 is shown outside the turbo decoder for ease of explanation, these two can be combined in implementation.

FIG. 4 is a block diagram of the turbo decoder of FIG. 3. As is evident, interleaver 402 and de-interleaver 401 exist between decoding circuitry 403 and decoding circuitry 404. Iterative decoding takes place as known in the art, however, unlike prior-art decoders, interleaver size K′ is related to K″ where K″ is from a set of sizes; wherein the set of sizes comprise K″=a^(p)×f, p_(min)≦p≦p_(max); f_(min)≦f≦f_(max), wherein a is an integer f is a continuous integer between f_(min) and f_(max) and p takes integer values between p_(min) and p_(max), a>1, p_(max)>p_(min), p_(min)>1. Parameter K_(filler) may or may not be needed by the turbo decoder 304, thus labeled as dashed line in FIG. 4.

As discussed above, in one embodiment K′=K″. In yet another embodiment K″=K″ when K″ is not a multiple of (2^(m)−1), otherwise using K′=K″+δ(K″) when K″ is a multiple of (2^(m)−1), wherein m is the memory length of a constituent convolutional encoder, and δ(K″) is a small positive or negative integer not equal to a multiple of (2^(m)−1) In one embodiment, m=3.

Interleaver 402 utilizes permutation π(i)=(iP₀+A+d(i))mod K′, where 0≦i≦K′−1 is the sequential index of the symbol positions after interleaving, π(i) is the symbol index before interleaving corresponding to position i, K′ is the interleaver size in symbols, P₀ is a number that is relatively prime to K′, A is a constant, C is a small number that divides K′ and 4(i) is a dither vector of the form d(i)=α(i mod C)+P₀×β(i mod C) where α(·) and β(·) are vectors each of length C, periodically applied for 0≦i≦K′−1. The values of K′, C, P₀, α(·) and β (·) are preferably taken from a row of Table 1. The deinterleaver 401 performs an inverse function of interleaver 402.

FIG. 5 is a flow chart showing operation of the transmitter 100. The logic flow begins at step 501 where circuitry 103 determines interleaver size K′ that is related to K″ where K″ is from a set of sizes; wherein the set of sizes comprise K″=a^(p)×f, p_(min)≦p≦p_(max); f_(min)≦f≦f_(max), wherein a is an integer, f is a continuous integer between f_(min) and f_(max), and p takes integer values between p_(min) and p_(max), a>1, p_(max)>p_(min), p_(min)>1. As discussed above, in one embodiment K′=K″. In yet another embodiment K′=K″ when K″ is not a multiple of (2^(m)−1), otherwise using K′=K″+δ(K″) when K″ is a multiple of (2^(m)−1), wherein m is the memory length of a constituent convolutional encoder, and δ(K″) is a small positive or negative integer not equal to a multiple of (2^(m)−1). In one embodiment, m=3.

At step 503 filler insertion circuitry 109 receives an information block of size K and pads the information block of size K into an input block u of size K′ and outputs the input block u. Interleaver 201 then interleaves he input block of size K′ (step 507) (preferably using a contention-free interleaver) and sends the interleaved block of size K′ to encoding circuitry 203 (step 509). Finally, at step 511, the original input block and interleaved input block are encoded.

As discussed above, the step of interleaving the input block comprises the step of using a permutation π(i)=(iP₀+A+d(i))mod K′, where 0≦i≦K′−1 is the sequential index of the bit positions after interleaving π(i) is the bit index before interleaving corresponding to position i, K′ is the interleaver size in bits, P₀ is a number that is relatively prime to K′, A is a constant, C is a small number that divides K′, and d(i) is a dither vector of the form d(i)=α(i mod C)+P₀×>β(i mod C) where α(·) and β(·) are vectors each of length C, periodically applied for 0≦i≦K′−1. The values of K′, C, P₀, α(·) and β (·) are preferably taken from Table 1.

FIG. 6 is a flow chart showing operation of the receiver of FIG. 3. The logic flow begins at step 601 where circuitry 306 determines the interleaver size K′. At Step 603, Circuitry 302 adds the filler bit information to the received signal vector, for example, if the filler bits and filler bit positions are known circuitry 302 may set to high magnitudes the Log Likelihood Ratios (LLRs) of those positions in the turbo decoder input. At step 607 turbo decoder decodes the decoder input block using interleaver and de-interleaver of size K′ and outputs an estimated û of the input block of length K′. At step 609 information block extraction circuitry 310 removes filler bits to obtain an estimate of the information block of length K. Finally at Step 611, the estimated information block is output.

While the invention has been particularly shown and described with reference to a particular embodiment, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention. In one example, the interleaver table may be enhanced further to handle special cases, including: (a) Using an additional set of interleaver sizes defined to cover any special block sizes that must be handled, e.g., without filler bits or with fewer filler bits. (b) The interleaver sizes can be slightly adjusted by adding or subtracting a small value from the semilog slice sizes. In another example, although the invention has been described above assuming binary-input turbo encoder, the same principle can be applied when the turbo encoder takes symbols as input. For example, a duo-binary turbo code takes a symbol of two binary bits at a time, and the turbo interleaver permutes symbols (further scrambling such as alternating the bits within a symbol may be performed). In such a case, the input block size is measured in symbols, and the interleaver size is equal to the number of symbols in the input block. In another example, although the above description assumes that the interleaver sizes and the interleaver parameters are stored in a look-up table, it is possible that they may be determined via other means such as algebraic calculation. In yet another example, although the above description assumes a turbo code, the method is also applicable to other FEC schemes including, for example low-density parity-check (LDPC) codes, Reed-Solomon (RS) Codes, etc. it is intended that such changes come within the scope of the following claims. 

1. A method for operating a turbo encoder, the method comprising the steps of: receiving an information block off size K; determining an interleaver size K′ that is related to K″ where K″ is from a set of sizes; wherein the set of sizes comprise K″=a^(p)×f, p_(min)≦p≦p_(max); f_(min)≦f≦f_(max), wherein a is i integer, f is a continuous integer between f_(min) and f_(max), and p takes integer values between p_(min) and p_(max), a>1, p_(max)>p_(min), p_(min)>1; padding the information block of size K into an input block of size K′; and interleaving the input block using an interleaver of size K′, encoding the original input block and the interleaved input block to obtain a codeword block; and transmitting the codeword block through the channel.
 2. The method of claim 1 wherein the step of determining an interleaver size K′ that is related to K″ comprise using K′=K″.
 3. The method of claim 1 wherein the step of determining an interleaver size K′ that is related to K″ comprise the step of using K′=K″ when K″ is not a multiple of (2^(m)−1), otherwise using K′=K″+δ(K″) when K″ is a multiple of (2^(m)−1), wherein m is a memory length of a constituent convolutional encoder, and δ(K″) is a small positive or negative integer not equal to a multiple of (2^(m)−1).
 4. The method of claim 3 wherein m=3.
 5. The method of claim 1 wherein f_(max)=a×f_(min)−1.
 6. The method of claim 1 wherein f_(max)=a×(f_(min)−1).
 7. The method of claim 1 wherein the step of interleaving the input block comprises the step of using a contention-free interleaver to interleave the input block.
 8. The method of claim 1 wherein the step of interleaving the input block comprises the step of using a permutation π(i)=(iP₀+A+d(i))mod K′, where 0≦i≦K′−1 is the sequential index of the symbol positions after interleaving, π(i) is the symbol index before interleaving corresponding to position i, K′ is the interleaver size in symbols, P₀ is a number that is relatively prime to K′, A is a constant, C is a small number that divides K′, and d(i) is a dither vector of the form d(i)=α(i mod C)+P₀×β(i mod C) where α(·) and β(·) are vectors each of length C, periodically applied for 0≦i≦K′−1.
 9. The method of claim 8 wherein K′=K″ when K″ is not a multiple of (2^(m)−1); otherwise K′=K″−dC or K′=K″+dC when K″ is a multiple of (2^(m)−1), wherein d is a small positive integer and d is not a multiple of (2^(m)−1), and m is a memory length of a constituent convolutional encoder.
 10. The method of claim 8 wherein the values of K′, C, P₀, α(·) and β (·) are taken from at least one row of the following table: Interleaver size K′ C P₀ α β 128 4 81 4 0 0 4 0 20 120 68 144 4 91 4 0 4 0 0 76 20 52 160 4 123 4 4 0 0 0 4 8 12 176 4 127 4 4 0 0 0 12 112 44 192 4 169 4 0 4 0 0 8 4 16 208 4 37 4 0 4 0 0 68 20 164 216 4 121 4 4 0 0 0 68 12 28 240 4 161 0 0 4 4 0 16 196 212 256 4 31 0 0 4 4 0 60 8 68 288 4 131 0 4 0 4 0 80 144 36 320 4 69 0 0 4 4 0 4 8 12 352 4 35 4 0 0 4 0 48 96 152 384 4 91 0 0 4 4 0 4 20 24 416 4 31 4 4 0 0 0 24 28 60 440 4 53 0 4 4 0 0 4 20 216 480 4 53 0 4 4 0 0 72 192 12 512 4 273 0 0 4 4 0 20 8 24 576 4 29 0 0 4 4 0 64 120 68 640 4 147 0 0 4 4 0 24 12 4 704 4 309 0 0 4 4 0 4 12 8 768 4 241 0 0 4 4 0 4 12 8 832 4 53 0 0 4 4 0 4 12 8 888 4 77 0 4 4 0 0 48 64 140 960 4 143 0 0 4 4 0 4 12 8 1024 8 245 8 0 8 8 0 0 8 0 0 8 40 16 96 80 56 88 1152 8 119 0 0 8 0 8 0 8 8 0 8 40 64 80 48 24 88 1280 8 897 0 0 8 0 8 0 8 8 0 8 96 88 32 16 48 40 1408 8 593 0 8 0 8 8 0 8 0 0 8 96 48 32 16 80 40 1536 8 1139 0 0 8 0 8 0 8 8 0 16 56 88 80 24 72 64 1664 8 1451 0 8 0 8 8 0 8 0 0 16 40 96 88 80 32 48 1776 8 115 8 0 0 0 0 8 8 8 0 88 56 40 152 120 128 200 1920 8 233 8 0 8 8 8 0 0 0 0 16 24 88 64 8 32 40 2048 8 77 0 0 8 8 0 8 8 0 0 64 136 160 48 192 24 120 2304 8 1631 0 0 8 0 8 0 8 8 0 24 80 40 16 96 64 32 2560 8 2249 0 0 8 0 8 0 8 8 0 8 72 40 88 48 32 96 2816 8 1235 8 8 0 0 0 8 0 8 0 16 88 96 56 24 48 64 3072 8 671 0 0 8 0 8 0 8 8 0 8 48 32 64 88 40 56 3328 8 1459 0 0 8 0 8 0 8 8 0 32 8 56 80 16 72 48 3568 8 147 0 0 8 0 8 0 8 8 0 72 64 48 88 8 184 248 3840 8 3721 8 0 8 8 8 0 0 0 0 16 48 24 8 32 40 88 4096 8 83 8 8 0 0 0 8 8 0 0 16 120 152 24 216 64 240 4608 8 181 0 8 0 8 0 8 8 0 0 32 176 216 136 64 224 248 5120 8 3629 0 8 0 8 8 0 8 0 0 16 40 96 88 80 32 48 5632 8 211 0 0 8 0 8 0 8 8 0 24 208 112 224 168 184 48 6144 8 4355 8 0 8 8 8 0 0 0 0 8 16 64 24 48 80 32


11. A method for operating a turbo encoder, the method comprising the steps of: receiving an input block of size K′; and encoding the input block using an interleaver of size K′ and a permutation π(i)=(iP₀+A+d(i))mod K′ where 0≦i≦K′−1 is a sequential index of the bit positions after interleaving, π (i) is a bit index before interleaving corresponding to position i, P₀ is a number that is relatively prime to K′, A is a constant equal to 3, C is a small number that divides K′ and d(i) is a dither vector of the form d(i)=α(i mod C)+P₀×δ(i mod C) where α(·) and β(·) are vectors each of length C, periodically applied for 0≦i≦K′−1 and wherein values of K′, C, P₀, α(·) and β (·) are taken from at least one row of the following table: Interleaver size K′ C P₀ α β 128 4 81 4 0 0 4 0 20 120 68 144 4 91 4 0 4 0 0 76 20 52 160 4 123 4 4 0 0 0 4 8 12 176 4 127 4 4 0 0 0 12 112 44 192 4 169 4 0 4 0 0 8 4 16 208 4 37 4 0 4 0 0 68 20 164 216 4 121 4 4 0 0 0 68 12 28 240 4 161 0 0 4 4 0 16 196 212 256 4 31 0 0 4 4 0 60 8 68 288 4 131 0 4 0 4 0 80 144 36 320 4 69 0 0 4 4 0 4 8 12 352 4 35 4 0 0 4 0 48 96 152 384 4 91 0 0 4 4 0 4 20 24 416 4 31 4 4 0 0 0 24 28 60 440 4 53 0 4 4 0 0 4 20 216 480 4 53 0 4 4 0 0 72 192 12 512 4 273 0 0 4 4 0 20 8 24 576 4 29 0 0 4 4 0 64 120 68 640 4 147 0 0 4 4 0 24 12 4 704 4 309 0 0 4 4 0 4 12 8 768 4 241 0 0 4 4 0 4 12 8 832 4 53 0 0 4 4 0 4 12 8 888 4 77 0 4 4 0 0 48 64 140 960 4 143 0 0 4 4 0 4 12 8 1024 8 245 8 0 8 8 0 0 8 0 0 8 40 16 96 80 56 88 1152 8 119 0 0 8 0 8 0 8 8 0 8 40 64 80 48 24 88 1280 8 897 0 0 8 0 8 0 8 8 0 8 96 88 32 16 48 40 1408 8 593 0 8 0 8 8 0 8 0 0 8 96 48 32 16 80 40 1536 8 1139 0 0 8 0 8 0 8 8 0 16 56 88 80 24 72 64 1664 8 1451 0 8 0 8 8 0 8 0 0 16 40 96 88 80 32 48 1776 8 115 8 0 0 0 0 8 8 8 0 88 56 40 152 120 128 200 1920 8 233 8 0 8 8 8 0 0 0 0 16 24 88 64 8 32 40 2048 8 77 0 0 8 8 0 8 8 0 0 64 136 160 48 192 24 120 2304 8 1631 0 0 8 0 8 0 8 8 0 24 80 40 16 96 64 32 2560 8 2249 0 0 8 0 8 0 8 8 0 8 72 40 88 48 32 96 2816 8 1235 8 8 0 0 0 8 0 8 0 16 88 96 56 24 48 64 3072 8 671 0 0 8 0 8 0 8 8 0 8 48 32 64 88 40 56 3328 8 1459 0 0 8 0 8 0 8 8 0 32 8 56 80 16 72 48 3568 8 147 0 0 8 0 8 0 8 8 0 72 64 48 88 8 184 248 3840 8 3721 8 0 8 8 8 0 0 0 0 16 48 24 8 32 40 88 4096 8 83 8 8 0 0 0 8 8 0 0 16 120 152 24 216 64 240 4608 8 181 0 8 0 8 0 8 8 0 0 32 176 216 136 64 224 248 5120 8 3629 0 8 0 8 8 0 8 0 0 16 40 96 88 80 32 48 5632 8 211 0 0 8 0 8 0 8 8 0 24 208 112 224 168 184 48 6144 8 4355 8 0 8 8 8 0 0 0 0 8 16 64 24 48 80 32


12. An apparatus for operating a turbo encoder, the apparatus comprising: interleaver size determination circuitry that determines an interleaver size K′ that is related to K″ where K″ is from a set of sizes; wherein the set of sizes comprise K″=a^(p)×f, p_(min)≦p≦p_(max); f_(min)≦f≦f_(max), wherein a is an integer, f is a continuous integer between f_(min), and f_(max), and p takes integer values between p_(min) and p_(max), a>1, p_(max)>p_(min), p_(min)>1; filler insertion circuitry receiving an information block of size K and padding the information block of size K into an input block of size K′; an interleaver interleaving the input block of size K′; and an encoder that encodes the original input block and the interleaved input block to obtain a codeword block.
 13. The apparatus of claim 12 where K′=K″.
 14. The apparatus of claim 12 where K′=K″ when K″ is not a multiple of (2^(m)−1), otherwise using K′=K″+δ(K″) when K″ is a multiple of (2^(m)−1), wherein m is the memory length of a constituent convolutional encoder, and δ(K″) is a small positive or negative integer not equal to a multiple of (2^(m)−1).
 15. The apparatus of claim 14 where m=3.
 16. The apparatus of claim 12 where f_(max)=a×f_(min)−1.
 17. The apparatus of claim 12 where f_(max)=a×(f_(min)−1).
 18. The apparatus of claim 12 wherein the interleaver uses a permutation π(i)=(iP₀+A+d(i))mod K′, where 0≦i≦K′−1 is a sequential index of a symbol positions after interleaving, π(i) is a symbol index before interleaving corresponding to position i, K′ is an interleaver size in symbols, P₀ is a number that is relatively prime to K′, A is a constant, C is a small number that divides K′, and d(i) is a dither vector of a form d(i)=α(i mod C)+P₀×β(i mod C) where α(·) and β(·) are vectors each of length C, periodically applied for 0≦i≦K′−1.
 19. The apparatus of claim 18 where the values of K′, C, P₀, α(·) and β (·) are taken from at least one row of the following table: Interleaver size K′ C P₀ α β 128 4 81 4 0 0 4 0 20 120 68 144 4 91 4 0 4 0 0 76 20 52 160 4 123 4 4 0 0 0 4 8 12 176 4 127 4 4 0 0 0 12 112 44 192 4 169 4 0 4 0 0 8 4 16 208 4 37 4 0 4 0 0 68 20 164 216 4 121 4 4 0 0 0 68 12 28 240 4 161 0 0 4 4 0 16 196 212 256 4 31 0 0 4 4 0 60 8 68 288 4 131 0 4 0 4 0 80 144 36 320 4 69 0 0 4 4 0 4 8 12 352 4 35 4 0 0 4 0 48 96 152 384 4 91 0 0 4 4 0 4 20 24 416 4 31 4 4 0 0 0 24 28 60 440 4 53 0 4 4 0 0 4 20 216 480 4 53 0 4 4 0 0 72 192 12 512 4 273 0 0 4 4 0 20 8 24 576 4 29 0 0 4 4 0 64 120 68 640 4 147 0 0 4 4 0 24 12 4 704 4 309 0 0 4 4 0 4 12 8 768 4 241 0 0 4 4 0 4 12 8 832 4 53 0 0 4 4 0 4 12 8 888 4 77 0 4 4 0 0 48 64 140 960 4 143 0 0 4 4 0 4 12 8 1024 8 245 8 0 8 8 0 0 8 0 0 8 40 16 96 80 56 88 1152 8 119 0 0 8 0 8 0 8 8 0 8 40 80 64 48 24 88 1280 8 897 0 0 8 0 8 0 8 8 0 8 96 88 32 16 48 40 1408 8 593 0 8 0 8 8 0 8 0 0 8 96 48 32 16 80 40 1536 8 1139 0 0 8 0 8 0 8 8 0 16 56 88 80 24 72 64 1664 8 1451 0 8 0 8 8 0 8 0 0 16 40 96 88 80 32 48 1776 8 115 8 0 0 0 0 8 8 8 0 88 56 40 152 120 128 200 1920 8 233 8 0 8 8 8 0 0 0 0 16 24 88 64 8 32 40 2048 8 77 0 0 8 8 0 8 8 0 0 64 136 160 48 192 24 120 2304 8 1631 0 0 8 0 8 0 8 8 0 24 80 40 16 96 64 32 2560 8 2249 0 0 8 0 8 0 8 8 0 8 72 40 88 48 32 96 2816 8 1235 8 8 0 0 0 8 0 8 0 16 88 96 56 24 48 64 3072 8 671 0 0 8 0 8 0 8 8 0 8 48 32 64 88 40 56 3328 8 1459 0 0 8 0 8 0 8 8 0 32 8 56 80 16 72 48 3568 8 147 0 0 8 0 8 0 8 8 0 72 64 48 88 8 184 248 3840 8 3721 8 0 8 8 8 0 0 0 0 16 48 24 8 32 40 88 4096 8 83 8 8 0 0 0 8 8 0 0 16 120 152 24 216 64 240 4608 8 181 0 8 0 8 0 8 8 0 0 32 176 216 136 64 224 248 5120 8 3629 0 8 0 8 8 0 8 0 0 16 40 96 88 80 32 48 5632 8 211 0 0 8 0 8 0 8 8 0 24 208 112 224 168 184 48 6144 8 4355 8 0 8 8 8 0 0 0 0 8 16 64 24 48 80 32


20. An apparatus for operating a turbo encoder, the apparatus comprising: a receiving circuitry that receives an input block of size K′; and an encoder that encodes an input block using an interleaver of size K′ and a permutation π(i)=(iP₀+A+d(i))mod K′ where 0≦i≦K′−1 is a sequential index of the bit positions after interleaving, π (i) is a bit index before interleaving corresponding to position i, P₀ is a number that is relatively prime to K′, A is a constant equal to 3, C is a small number that divides K′, and d(i) is a dither vector of the form d(i)=α(i mod C)+P₀×β(i mod C) where α(·) and β(·) are vectors each of length C, periodically applied for 0≦i≦K′−1 and wherein values of K′, C, P₀, α(·) and β (·) are taken from at least one row of the following table: Interleaver size K′ C P₀ α β 128 4 81 4 0 0 4 0 20 120 68 144 4 91 4 0 4 0 0 76 20 52 160 4 123 4 4 0 0 0 4 8 12 176 4 127 4 4 0 0 0 12 112 44 192 4 169 4 0 4 0 0 8 4 16 208 4 37 4 0 4 0 0 68 20 164 216 4 121 4 4 0 0 0 68 12 28 240 4 161 0 0 4 4 0 16 196 212 256 4 31 0 0 4 4 0 60 8 68 288 4 131 0 4 0 4 0 80 144 36 320 4 69 0 0 4 4 0 4 8 12 352 4 35 4 0 0 4 0 48 96 152 384 4 91 0 0 4 4 0 4 20 24 416 4 31 4 4 0 0 0 24 28 60 440 4 53 0 4 4 0 0 4 20 216 480 4 53 0 4 4 0 0 72 192 12 512 4 273 0 0 4 4 0 20 8 24 576 4 29 0 0 4 4 0 64 120 68 640 4 147 0 0 4 4 0 24 12 4 704 4 309 0 0 4 4 0 4 12 8 768 4 241 0 0 4 4 0 4 12 8 832 4 53 0 0 4 4 0 4 12 8 888 4 77 0 4 4 0 0 48 64 140 960 4 143 0 0 4 4 0 4 12 8 1024 8 245 8 0 8 8 0 0 8 0 0 8 40 16 96 80 56 88 1152 8 119 0 0 8 0 8 0 8 8 0 8 40 64 80 48 24 88 1280 8 897 0 0 8 0 8 0 8 8 0 8 96 88 32 16 48 40 1408 8 593 0 8 0 8 8 0 8 0 0 8 96 48 32 16 80 40 1536 8 1139 0 0 8 0 8 0 8 8 0 16 56 88 80 24 72 64 1664 8 1451 0 8 0 8 8 0 8 0 0 16 40 96 88 80 32 48 1776 8 115 8 0 0 0 0 8 8 8 0 88 56 40 152 120 128 200 1920 8 233 8 0 8 8 8 0 0 0 0 16 24 88 64 8 32 40 2048 8 77 0 0 8 8 0 8 8 0 0 64 136 160 48 192 24 120 2304 8 1631 0 0 8 0 8 0 8 8 0 24 80 40 16 96 64 32 2560 8 2249 0 0 8 0 8 0 8 8 0 8 72 40 88 48 32 96 2816 8 1235 8 8 0 0 0 8 0 8 0 16 88 96 56 24 48 64 3072 8 671 0 0 8 0 8 0 8 8 0 8 48 32 64 88 40 56 3328 8 1459 0 0 8 0 8 0 8 8 0 32 8 56 80 16 72 48 3568 8 147 0 0 8 0 8 0 8 8 0 72 64 48 88 8 184 248 3840 8 3721 8 0 8 8 8 0 0 0 0 16 48 24 8 32 40 88 4096 8 83 8 8 0 0 0 8 8 0 0 16 120 152 24 216 64 240 4608 8 181 0 8 0 8 0 8 8 0 0 32 176 216 136 64 224 248 5120 8 3629 0 8 0 8 8 0 8 0 0 16 40 96 88 80 32 48 5632 8 211 0 0 8 0 8 0 8 8 0 24 208 112 224 168 184 48 6144 8 4355 8 0 8 8 8 0 0 0 0 8 16 64 24 48 80 32 